## Hamilton-Jacobi equations with state constraints.(English)Zbl 0702.49019

Summary: We consider Hamilton-Jacobi equations of the form $$H(x,u,\nabla u)=0$$, $$x\in \Omega$$, where $$\Omega$$ is a bounded open subset of $$R^ n$$, H is a given continuous real-valued function of (x,s,p)$$\in \Omega \times R\times R^ n$$ and $$\nabla u$$ is the gradient of the unknown function u. We are interested in particular solutions of the above equation which are required to be supersolutions, in a suitable weak sense, of the same equation up to the boundary of $$\Omega$$.
This requirement plays the role of a boundary condition. The main motivation for this kind of solution comes from deterministic optimal control and differential games problems with constraints on the state of the system, as well from related questions in constrained geodesics.

### MSC:

 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 49L20 Dynamic programming in optimal control and differential games

### Keywords:

viscosity super solutions; supersolutions
Full Text:

### References:

 [1] G. Barles, Existence results for first order Hamilton Jacobi equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 5, 325 – 340 (English, with French summary). · Zbl 0574.70019 [2] G. Barles, Remarques sur des résultats d’existence pour les équations de Hamilton-Jacobi du premier ordre, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 1, 21 – 32 (French, with English summary). · Zbl 0573.35010 [3] G. Barles and P.-L. Lions, in preparation. [4] A. Bensoussan, Méthodes de perturbation en controle optimal (to appear). [5] I. Capuzzo-Dolcetta and M. G. Garroni, Oblique derivative problems and invariant measures, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), no. 4, 689 – 720. · Zbl 0635.35020 [6] M. G. Crandall, L. C. Evans, and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), no. 2, 487 – 502. · Zbl 0543.35011 [7] M. G. Crandall, H. Ishii and P.-L. Lions, Uniqueness of viscosity solutions revisited. · Zbl 0644.35016 [8] Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1 – 42. · Zbl 0599.35024 [9] -, On existence and uniqueness of solutions of Hamilton-Jacobi equations, Nonlinear Anal. (1985). [10] -, Hamilton-Jacobi equations in infinite dimensions. Parts I, II, III, J. Funct. Anal. 62 (1985), 379-396; 65 (1986), 368-495; 68 (1986), 214-247; announced in C. R. Acad Sci. Paris Sér. I Math. 300 (1985), 67-70. [11] Michael G. Crandall and Pierre-Louis Lions, Remarks on the existence and uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations, Illinois J. Math. 31 (1987), no. 4, 665 – 688. · Zbl 0678.35009 [12] M. G. Crandall, P.-L. Lions and P. E. Souganidis, in preparation. [13] Michael G. Crandall and Richard Newcomb, Viscosity solutions of Hamilton-Jacobi equations at the boundary, Proc. Amer. Math. Soc. 94 (1985), no. 2, 283 – 290. · Zbl 0575.35008 [14] Francis Gimbert, Problèmes de Neumann quasilinéaires, J. Funct. Anal. 62 (1985), no. 1, 65 – 72 (French). · Zbl 0579.35028 [15] R. Gonzalez and E. Rofman, On deterministic control problems: an approximation procedure for the optimal cost, Parts I and II, SIAM J. Control Optim. 23 (1985). · Zbl 0563.49025 [16] Hitoshi Ishii, Perron’s method for Hamilton-Jacobi equations, Duke Math. J. 55 (1987), no. 2, 369 – 384. · Zbl 0697.35030 [17] Hitoshi Ishii, Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations, Indiana Univ. Math. J. 33 (1984), no. 5, 721 – 748. · Zbl 0551.49016 [18] Hitoshi Ishii, Remarks on existence of viscosity solutions of Hamilton-Jacobi equations, Bull. Fac. Sci. Engrg. Chuo Univ. 26 (1983), 5 – 24. · Zbl 0546.35042 [19] -, Existence and uniqueness of solutions of Hamilton-Jacobi equations, preprint. [20] -, A simple, direct proof of uniqueness for solutions of the Hamilton-Jacobi equations of Eikonal type, preprint. · Zbl 0644.35017 [21] R. Jensen, work in preparation and personal communication. [22] Jean-Michel Lasry and Pierre-Louis Lions, Équations elliptiques non linéaires avec conditions aux limites infinies et contrôle stochastique avec contraintes d’état, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 7, 213 – 216 (French, with English summary). · Zbl 0568.35042 [23] Pierre-Louis Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. · Zbl 0497.35001 [24] P.-L. Lions, Optimal control and viscosity solutions, Recent mathematical methods in dynamic programming (Rome, 1984) Lecture Notes in Math., vol. 1119, Springer, Berlin, 1985, pp. 94 – 112. [25] P.-L. Lions, Neumann type boundary conditions for Hamilton-Jacobi equations, Duke Math. J. 52 (1985), no. 4, 793 – 820. · Zbl 0599.35025 [26] P.-L. Lions, Quelques remarques sur les problèmes elliptiques quasilinéaires du second ordre, J. Analyse Math. 45 (1985), 234 – 254 (French). · Zbl 0614.35034 [27] P.-L. Lions, G. Papanicolau and S. R. S. Varadhan, in preparation. [28] P.-L. Lions and B. Perthame, Quasivariational inequalities and ergodic impulse control, SIAM J. Control Optim. 24 (1986), no. 4, 604 – 615. [29] B. Perthame and R. Sanders, The Neumann problem for fully nonlinear second order singular pertubation problems, M.R.C. Technical Summary Report, Univ. of Wisconsin-Madison, 1986. [30] Maurice Robin, Long-term average cost control problems for continuous time Markov processes: a survey, Acta Appl. Math. 1 (1983), no. 3, 281 – 299. · Zbl 0531.93068 [31] Maurice Robin, On some impulse control problems with long run average cost, SIAM J. Control Optim. 19 (1981), no. 3, 333 – 358. · Zbl 0461.93062 [32] Halil Mete Soner, Optimal control with state-space constraint. I, SIAM J. Control Optim. 24 (1986), no. 3, 552 – 561. · Zbl 0597.49023 [33] Panagiotis E. Souganidis, Existence of viscosity solutions of Hamilton-Jacobi equations, J. Differential Equations 56 (1985), no. 3, 345 – 390. · Zbl 0506.35020 [34] I. Capuzzo-Dolcetta and J.-L. Menaldi, On the deterministic optimal stopping time problem in the ergodic case, Theory and applications of nonlinear control systems (Stockholm, 1985) North-Holland, Amsterdam, 1986, pp. 453 – 460.
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